This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.

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The electronic properties of graphene

We investigate electronic transport in lithographically patterned graphene ribbon structures where the lateral confinement of charge carriers creates an energy gap near the charge neutrality point. Individual graphene layers are contacted with metal electrodes and patterned into ribbons of varying widths and different crystallographic orientations. The temperature dependent conductance measurements show larger energy gaps opening for narrower ribbons. The sizes of these energy gaps are investigated by measuring the conductance in the nonlinear response regime at low temperatures. We find that the energy gap scales inversely with the ribbon width, thus demonstrating the ability to engineer the band gap of graphene nanostructures by lithographic processes.

Energy band-gap engineering of graphene nanoribbons.

Raman spectroscopy of carbon nanotubes

Electrical current can be completely spin polarized in a class of materials known as half-metals, as a result of the coexistence of metallic nature for electrons with one spin orientation and insulating nature for electrons with the other. Such asymmetric electronic states for the different spins have been predicted for some ferromagnetic metals--for example, the Heusler compounds--and were first observed in a manganese perovskite. In view of the potential for use of this property in realizing spin-based electronics, substantial efforts have been made to search for half-metallic materials. However, organic materials have hardly been investigated in this context even though carbon-based nanostructures hold significant promise for future electronic devices. Here we predict half-metallicity in nanometre-scale graphene ribbons by using first-principles calculations. We show that this phenomenon is realizable if in-plane homogeneous electric fields are applied across the zigzag-shaped edges of the graphene nanoribbons, and that their magnetic properties can be controlled by the external electric fields. The results are not only of scientific interest in the interplay between electric fields and electronic spin degree of freedom in solids but may also open a new path to explore spintronics at the nanometre scale, based on graphene.

https://arxiv.org/pdf/cond-mat/0611600.pdf

Half-metallic graphene nanoribbons

Every few years, a new material with unique properties emerges and fascinates the scientific community, typical recent examples being high-temperature superconductors and carbon nanotubes. Graphene is the latest sensation with unusual properties, such as half-integer quantum Hall effect and ballistic electron transport. This two-dimensional material which is the parent of all graphitic carbon forms is strictly expected to comprise a single layer, but there is considerable interest in investigating two-layer and few-layer graphenes as well. Synthesis and characterization of graphenes pose challenges, but there has been considerable progress in the last year or so. Herein, we present the status of graphene research which includes aspects related to synthesis, characterization, structure, and properties.

Graphene: The New Two-Dimensional Nanomaterial

Electronic and magnetic properties of ribbon-shaped nanographite systems with zigzag and armchair edges in a magnetic field are investigated by using a tight-binding model. One of the most remarkable features of these systems is the appearance of edge states, strongly localized near zigzag edges. The edge state in a magnetic field, generating a rational fraction of the magnetic flux ( f5 p/q) in each hexagonal plaquette of the graphite plane, behaves like a zero-field edge state with q internal degrees of freedom. The orbital diamagnetic susceptibility strongly depends on the edge shapes. The reason is found in the analysis of the ring currents, which are very sensitive to the lattice topology near the edge. Moreover, the orbital diamagnetic susceptibility is scaled as a function of the temperature, Fermi energy, and ribbon width. Because the edge states lead to a sharp peak in the density of states at the Fermi level, the graphite ribbons with zigzag edges show Curie-like temperature dependence of the Pauli paramagnetic susceptibility. Hence, there is a crossover from hightemperature diamagnetic to low-temperature paramagnetic behavior in the magnetic susceptibility of nanographite ribbons with zigzag edges. @S0163-1829~99!02111-6#

https://arxiv.org/pdf/cond-mat/9809260.pdf

Electronic and magnetic properties of nanographite ribbons

The π-electron states of carbon nanotubes (CN's) in magnetic fields are calculated in the effective-mass theory. A sensitive change of CN from metal to semiconductor depending on its structure is well reproduced. The band gap is inversely proportional to the tube diameter and exhibits a drastic change as a function of magnetic flux passing through the cylinder with period c h / e due to the Aharonov-Bohm effect. In a magnetic field perpendicular to the tube axis, the band-gap is reduced strongly and the energy spectra approach those of a graphite sheet.

Electronic States of Carbon Nanotubes

Optical absorption spectra are calculated in carbon nanotubes in the presence of a magnetic flux parallel to the tube axis. A drastic change in the band gap manifests itself in optical spectra for light polarization parallel to the axis. In the case of perpendicular polarization, the absorption is suppressed by a large depolarization effect.

Aharonov-Bohm effect in carbon nanotubes

Magnetic properties of carbon nanotubes (CN's) are calculated in the effective-mass theory. The magnetic moment in a magnetic flux parallel to the tube axis oscillates as a function of the flux with period determined by the flux quantum and phase depending on whether CN is metallic or semiconducting. For a magnetic field perpendicular to the axis, CN's exhibit a large diamagnetism independent of metallic or semiconducting. The magnetic moment induced by a perpendicular field is about three order of magnitude as large as that by a parallel field.

Magnetic Properties of Carbon Nanotubes

Electronic states of carbon nanotubes in a magnetic field perpendicular to the axis are studied in a tight-binding model and the validity of k · p theory is examined. The energy bands near the Fermi level calculated in the tight-binding model agree with those of k · p in a wide magnetic-field range except in extremely high fields where the flux passing through the unit cell of a two-dimensional graphite is comparable to the flux quantum, i.e. up to ∼4000 T. A slight deviation from the k · p result in weak magnetic fields can be explained by the inclusion of higher order terms in the k · p expansion leading to trigonal warping of bands.

Hiroshi Ajiki

Papers

Electronic and magnetic properties of nanographite ribbons

Electronic States of Carbon Nanotubes

Aharonov-Bohm effect in carbon nanotubes

Magnetic Properties of Carbon Nanotubes

Energy bands of carbon nanotubes in magnetic fields